Your unit does know and has informed you correctly. You are already a fully trained soldier as TSC(B) is the non-infantry equivalent of TSC(Inf), so you only need to complete your trade training, which is the IMP(V) course.

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Wrong.

You have not completed CIC so you have not finished the cycle.

You will now need to complete TSC(B) to complete the corps training cycle, then go onto your trade training.

Also depending when you finished TSC(Inf) you may have to reattend a refresher before attending TSC(B), or even start all over again!